First, try connecting your iOS device to a Mac or PC running iTunes using a USB cable. A device list will appear in the top left corner of iTunes, with each device having its respective icon. Select your connected device from this list.
If the iOS device you connected to the computer does not appear on this list, perform a force restart. To do so, press and hold the sleep/wake and home buttons simultaneously for at least 10 seconds. The force restart is successful when the Apple logo appears on the screen.
From here, the next step is to restore the backup. If you previously made an iTunes backup before you attempted to upgrade your device to iOS 9, you can use that backup to restore your device. Otherwise, click the Back Up Now button to make a backup of your device in iTunes and restore from that backup. A separate page on Apple's support website explains how to create a backup in iTunes.
This should do the trick, but it may not work for all iOS users. Apple is expected to roll out a permanent fix in a future iOS update, but for now it doesn't offer any alternative solution.